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<H1><CENTER>Superposition of Waves</CENTER></H1>
The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time.  The waves pass through each other without being disturbed.  The net displacement of the medium at any point in space or time, is simply the sum of the individual wave dispacements.    This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.  
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<H3>Two guassian waves travelling in opposite directions</H3>
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The movie at left shows two gaussian wave pulses are travelling on a string, one is moving to the right, the other is moving to the left.  They pass through each other without being disturbed, and the net displacement is the sum of the two individual displacements.  It should also be mentioned that this string is nondispersive (all frequencies travel at the same speed) since the gaussian wave pulses do not change their shape as they propagate.  If the medium was dispersive, then the waves would change their shape.
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<H2>Two sine waves travelling in the same direction: Constructive and Destructive Interference</H2>
Two waves (with the same amplitude, frequency, and wavelength) are travelling in the same direction on a string.  Using the principle of superposition, the resulting string displacement may be written as:<p>
<center><img src="Eq1.gif"></center><br>
which is a travelling wave whose amplitude depends on the phase (phi).  When the two waves are <b>in-phase</b> (phi=0), they interfere <b>constructively</b> and the result has twice the amplitude of the individual waves.  When the two waves have <b>opposite-phase</b> (phi=180), they interfere <b>destructively</b> and cancel each other out.    
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The animation at left shows two sinusoidal waves travelling in the same direction.  The phase difference between the two waves varies increases with time so that the effects of both constructive and destructive interference may be seen.  First of all, notice that the sum wave (in blue) is a travelling wave which moves from left to right.  When the two gray waves are in phase the result is large amplitude.  When the two gray waves become out of phase the sum wave is zero.
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<H2>Two sine waves travelling in opposite directions create a standing wave</H2>
A <b>travelling wave</b> moves from one place to another, whereas a <b>standing wave</b> appears to stand still, vibrating in place.   Two waves (with the same amplitude, frequency, and wavelength) are travelling in opposite directions on a string.  Using the principle of superposition, the resulting string displacement may be written as:<p>
<center><img src="Eq2.gif"></center><br>
This wave is no longer a travelling wave because the position and time dependence have been separated.  The displacement of the string as a function of position has an amplitude of <i>2y<sub>m</sub></i>sin<i> kx</i>. This amplitude does not travel along the string, but stands still and oscillates up and down according to cos<i> wt</i>.  Characteristic of standing waves are locations with maximum displacement (<b>antinodes</b>) and locations with zero displacement (<b>nodes</b>). 
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The movie at left shows how a standing wave may be created from two travelling waves.  If two sinusoidal waves having the same frequency (wavelength) and the same amplitude are travelling in opposite directions in the same medium then, using superposition, the net displacement of the medium is the sum of the two waves.  As the movie shows, when the two waves are 180&deg; out-of-phase with each other they cancel, and when they are in-phase with each other they add together.  As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left.  I have placed two dots on the string, one at an antinode and one at a node.  Which is which?
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<H2>Two sine waves with different frequencies: Beats</H2>
Two waves of equal amplitude are travelling in the same direction. The two waves have different frequencies and wavelengths, but they both travel with the same wave speed.  Using the principle of superposition, the resulting particle displacement may be written as:<p>
<center><img src="Eq3.gif"></center><br>
This resulting particle motion is the product of <b>two</b> travelling waves.  One part is a sine wave which oscillates with the average frequency <i>f</i> = &frac12;(<i>f<sub>1</sub> + f<sub>2</sub></i>).  This is the frequency which is perceived by a listener.  The other part is a cosine wave which oscillates with the difference frequency <i>f</i> = &frac12;(<i>f<sub>1</sub> - f<sub>2</sub></i>).   This term controls the amplitude "envelope" of the wave and causes the perception of "beats".  The beat frequency is actually twice the difference frequency, <i>f</i><sub>beat</sub> = (<i>f<sub>1</sub> - f<sub>2</sub></i>).  <p>
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In the movie at left two waves with slightly different frequencies are travelling to the right.  The resulting wave travels in the same direction and with the same speed as the two component waves.  The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.  
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